A hypothetical space elevator question.

[Pattonias]

I am a fledgling undergrad with a potential mechanical engineering degree in my future.

I just have a hypothetical question about the space elevator that yall may be able to help me with. I have not taken my physics courses yet so forgive my limited working knowledge of the material. If you can make reference to any math you use, let me know and I will study it on my own to better understand.

If you have a space elevator with required x height, how long would it have to be in order for the climbing car to begin to travel up/outward due to the inertia of the spinning earth.

I know that you would have to overcome gravity. Would the tether be so long that you couldn't use the inertia as a method of moving the payload?

Could you use the tether as a guide for a blimp or weather balloon until the balloon reached a maximum altitude and then propel (using rocket or mechanical device) the payload until the inertia took over?

 

[DaveC426913]

The rising payload experiences the downward (inward) force of gravity until it reaches the geostationary point - 36,000km, where it reaches zero. After that, the force increases outward.

One of the really cool things about space elevators (or "beanstalks") is that you can use them to launch vehicles with some residual velocity for "free".

 

[Pattonias]

Cool, that actually makes a lot of sense. So a geostationary orbit is the neutral spot so to speak.

 

[DaveC426913]

Cool, that actually makes a lot of sense. So a geostationary orbit is the neutral spot so to speak.

Technically, the whole elevator is geostationary by definition. A satellite at 36,000km altitude is in geostationary orbit and would float next to that point on the cable. So if you reached out from the satellite and jumped into the cable, you'd be weightless.

Any spot below that on the cable is moving too slow to keep in a circular orbit, so it experiences a correspending increase in gravity. Any spot above that on the cable is moving too fast to keep in a circular orbit, so it experiences a correspending acceleration in velocity by being flung outward due to Earth's (and cable's) rotation.

 

[Pattonias]

Could you place a counter weight beyond the satellite and use it to haul up the cargo without having to use rockets or mechanical climber? Would it be easier to haul the counter weight back toward the satellite that it would be to climb up manually. Essentially like a normal elevator using the inertia of the counterweight instead of gravity to ease the burden on the climbing mechanism. A machine in orbit would be able to operate using solar power to power the climb instead of relying on the mechanisms power. This could lighten the load of the climber as it would not need the climbing machinery. It should also reduce the stress on the cable.

 

[DaveC426913]

Could you place a counter weight beyond the satellite and use it to haul up the cargo without having to use rockets or mechanical climber? Would it be easier to haul the counter weight back toward the satellite that it would be to climb up manually. Essentially like a normal elevator using the inertia of the counterweight instead of gravity to ease the burden on the climbing mechanism. A machine in orbit would be able to operate using solar power to power the climb instead of relying on the mechanisms power. This could lighten the load of the climber as it would not need the climbing machinery. It should also reduce the stress on the cable.

Yes. One you get it going, with payload going both ways, it cancels out. You can use the energy of captured incoming payloads to haul loads up the cable.

You start to see why it is such a compelling solution...

 

[spacester]

Yes. One you get it going, with payload going both ways, it cancels out. You can use the energy of captured incoming payloads to haul loads up the cable.

That is slightly contrary to my understanding. I'll attempt to clarify.

The cable (ribbon, actually) is held taut by the force of gravity below the center of mass of the entire elevator satellite. It is held taut above that point by "centrifugal force" (centripetal acceleration). A mid-way station need not be right at the GEO height, just as long as the center of mass of the system is.

A elevator car needs to climb the ribbon, there is no pull rope after all. When the car climbs up, if nothing else changes at the same time, the center of mass of the system will be lower than it should be for a stable orbit. So when the car gets to GEO altitude, it has put the system out of whack. This can be understood in terms of conservation of angular momentum or by looking at the energy picture.

When a car climbs up, it is contributing energy to the system in terms of potential energy from going straight up. But this is a fraction (um, 1/3rd IIRC) of the energy needed to put that car in orbit. Yet it is in fact in orbit.

So to put the system back in whack :-) you need to add energy. In practice, this simple elevator would fire a rocket to boost the orbit back up where it needs to be. But this is where the nifty energy trade-off comes in.

If you have an elevator car climb from the outer tip, way up past GEO, and climb to the midway station, it actually is contributing to the energy picture by climbing "up" just like the lower car does. But as it comes to lower altitudes, the energy it had from being up in a heliocentric orbit and not bound to a gravitation well gets pumped into the orbital energy of the elevator system.

The outer car coming in binds its energy to the elevator and raises the orbit and the lower car climbing up liberates energy from the orbit and lowers it. So hey presto you do both at the same time and they cancel out.

But the car still has to find a way to climb up, and that's a lot of potential energy to find a source for.

There are operational tricks that let you do some interesting things in terms of down-mass supplying most of the energy to operate a space elevator. I like the parallel to the Panama Canal, where the rainforest rainfall provides the energy to run the whole canal.

That's my understanding of it, anyway.